I recently asked some friends a rather lame question (due to Brian Greene):
Suppose you are on Deal or No Deal, and you have just won a prize consisting of an infinite collection of envelopes:
The first envelope contains $1, the second envelope contains $2, the third envelope contains $3,...., the Nth envelope contains $N, .... (ad infinitum). Now, the host offers to double the amount of money in each envelope. Deal or No Deal?
Wiseguys who have taken some calculus or analysis will quickly point out that both offers are worth Infinity dollars, so there's no difference in the two offers.
Guys who realize that they only live for 80 years, or have only storage space for 2,000,000 envelopes, will shout "Deal!", and then grab as many envelopes as they can, whose contents have now been doubled by the host.
Immortal types will realize that the original offer already contains the doubled-up offer, plus other envelopes with odd numbers of dollars. Thus, after declaring "No Deal!", they will proceed to pocket the proceedings from the envelopes until the End of Time.
Mathematicians and string theorists, after assessing their options using various arcane tricks, will depart with "No thanks, keep your money Sir!"
It's not because they are weirdos who couldn't give two hoots about becoming insanely rich. More probably, they are too poor to accept either offer...
This is because 1+2+3+4+.... is really equal to
Suppose you are on Deal or No Deal, and you have just won a prize consisting of an infinite collection of envelopes:
The first envelope contains $1, the second envelope contains $2, the third envelope contains $3,...., the Nth envelope contains $N, .... (ad infinitum). Now, the host offers to double the amount of money in each envelope. Deal or No Deal?
Wiseguys who have taken some calculus or analysis will quickly point out that both offers are worth Infinity dollars, so there's no difference in the two offers.
Guys who realize that they only live for 80 years, or have only storage space for 2,000,000 envelopes, will shout "Deal!", and then grab as many envelopes as they can, whose contents have now been doubled by the host.
Immortal types will realize that the original offer already contains the doubled-up offer, plus other envelopes with odd numbers of dollars. Thus, after declaring "No Deal!", they will proceed to pocket the proceedings from the envelopes until the End of Time.
Mathematicians and string theorists, after assessing their options using various arcane tricks, will depart with "No thanks, keep your money Sir!"
It's not because they are weirdos who couldn't give two hoots about becoming insanely rich. More probably, they are too poor to accept either offer...
This is because 1+2+3+4+.... is really equal to
-1/12.(Terry Tao says so! See his Eqn (5), or alternatively, Wikipedia.)
